The Beltrami-Klein model has the advantage that lines in the model resemble Euclidean lines; however, it has the drawback that it is not angle preserving. That is, the Euclidean measure of an angle within the model is not necessarily the angle measure in hyperbolic geometry.

Some points outside of the Beltrami-Klein model are important for constructions within the model. The following is an example of such:

Let ℓnormal-ℓ\ell be a line in the Beltrami-Klein model that is not a diameter of the circle. The pole of ℓnormal-ℓ\ell is the intersection of the Euclidean lines that are tangent to the circle at the endpoints of ℓnormal-ℓ\ell.

ℓnormal-ℓ\ellP⁢(ℓ)Pnormal-ℓP(\ell)...

Poles are important for the following reason: Given a line ℓnormal-ℓ\ell that is not a diameter of the Beltrami-Klein model, one constructs a line perpendicular to ℓnormal-ℓ\ell by considering Euclidean lines passing throughP⁢(ℓ)Pnormal-ℓP(\ell). Thus, given two disjointly parallel linesℓnormal-ℓ\ell and mmm that are not diameters of the Beltrami-Klein model, one constructs their common perpendicular by connecting their poles.

ℓnormal-ℓ\ellmmmP⁢(ℓ)Pnormal-ℓP(\ell)P⁢(m)PmP(m)...nnn

In the above picture, nnn is the common perpendicular of ℓnormal-ℓ\ell and mmm.